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I don't understand how to proof the eigenvalues of rotation matrix using geometric argument only. I know how to do it with determinant but what about the geometric way? What is the meaning of complex number as eigenvalues?

For example, find the eigenvalues for R geometrically:

$$ R = \pmatrix{\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}} $$

(image link)

Ben Grossmann
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Lily Nguyen
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  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Nov 28 '22 at 23:54
  • If you're familiar with the isomorphism between the complex numbers and certain $2 \times 2$ real matrices you can show that the eigenvalues act on the argand plane in the same manner that the matrix acts on $\mathbb{R}^2$. Basically the polar form of the eigenvalue gives you the argument of the rotation. – CyclotomicField Nov 29 '22 at 02:00

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